POLYNOMIAL SOLUTIONS TO PIEZOELECTRIC BEAMS (Ⅱ)——ANALYTICAL SOLUTIONS TO TYPICAL PROBLEMS

IntroductionThis paper is a continuation of Ref.[1],in which a series of orthotropic piezoelectricplane problems was solved and the corresponding exact solutions were obtained with the trial-and-error method,on the basis of the general solution expressed …     

Nonlinear problems of shell theory and their solution methods (review)

Conclusions In the present review we considered several problems of nonlinear shell theory, their deformation and stability. Speciailsts in many areas have devoted much attention recently to problems of the nonlinear behavior of evolutionary systems. New scientific trends related to the study of stability problems have been generated. They can be called the Hopf bifurcation theory; or, the theory of attractors, referring to the study of unstable trajectory behavior in phase spaces; or, catastrophe theory, applied in solving multiparametric stability problems; or, the theory of phase transitions and critical phenomena, developed in special areas of physics; or, a theory considering nonequilibrium phase transitions and self-organization complex spatial and temporal ordered structures (synergetics).The use of methods developed by these scientific trends in shell theory enables deep understanding of effects of stability loss, in both equilibrium and motion, and helps to create investigation methods.Institute of Mechanics, Ukrainian Academy of Sciences, Kiev. Kiev Institute of Structural Engineering. Translated from Prikladnaya Mekhanika, Vol. 27, No. 10, pp. 3–23, October, 1991.     

Existence and stability of solutions to inverse variational inequality problems

In this paper, two new existence theorems of solutions to inverse variational and quasi-variational inequality problems are proved using the Fan-Knaster-KuratowskiMazurkiewicz(KKM) theorem and the Kakutani-Fan-Glicksberg fixed point theorem.Upper semicontinuity and lower semicontinuity of the solution mapping and the approximate solution mapping to the parametric inverse variational inequality problem are also discussed under some suitable conditions. An application to a road pricing problem is given.     

General theorems and fundamental solutions in the dynamical theory of mixtures

This paper is concerned with the dynamic theory of a binary mixture consisting of a gas and an elastic solid. First, we derive general theorems (reciprocal theorem, uniqueness theorem, minimum principle). The, we establish a Galerkin representation and the fundamental solutions in the case of steady vibrations.Work performed under the auspices of G.N.F.M. of the Italian Research Council (C.N.R.), with the grant 60% M.U.R.S.T. (ITALY).     

Analytical solutions of peristatic and peridynamic problems for a 1D infinite rod

Peridynamics is a nonlocal theory of continuum mechanics, which was developed by Silling (2000). Since then peridynamics has been applied to a variety of solid mechanics problems ranging from fracture, damage, failure to wave propagation, buckling, and detonation physics. Since the governing equation of peridynamics is an integro-differential equation, most of the treatment in the literature is often numerical. However, the analytical treatment is very important for the development of the peridynamic theory, which is continually developing at the present time. In this paper, peristatic and peridynamic problems for a 1D infinite rod are analytically investigated. We have developed a method to obtain a valid analytical solution starting from a formal analytical solution, which may be divergent. The primary contribution of the present paper is a systematic analytical treatment of peristatic and peridynamic problems for a 1D infinite rod. Additionally, dispersion curves and group velocities for the materials with three different micromoduli are also studied. It is found from the study that some peridynamic materials can have negative group velocities in certain regions of wavenumber. This indicates that peridynamics can be used for modeling certain types of dispersive media with anomalous dispersion such as the one discussed by Mobley (2007).     

Elastoplastic problems in the uniaxially strained state approximation. Analytical and numerical solutions

A detailed study of shock and unloading waves is given in the uniaxially strained state approximation for elastoplastic problems. A numerical algorithm is briefly described. The TIS-1D software complex is verified in the case of one-dimensional problems. This software complex is based on the method of separation into physical processes and on the finite volume method using moving Eulerian grids.     

Energy and variational principles for generalized (gyroscopic) conservative problems

The study of the local stability of the equilibrium of an elastic body subjected to conservative and gyroscopic forces, which was initiated in [6] is extended. Two variational principles are established in actual and equivalent energy terms respectively. These principles, together with the stability criterion formulated in [6], form a generalized version of (and reduce to) the classical energy principles governing the stability of conservative non-gyroscopic systems, and are applied to the example of a loaded (or unloaded) elastic shaft under constant rotation. In addition, a necessary and sufficient condition for the applied forces to be gyroscopic or conservative is obtained explicitly as a function of the change in force components due to a small disturbance superimposed on the equilibrium state.     

Hybrid projection method for generalized mixed equilibrium problems,variational inequality problems,and fixed point problems in Banach spaces

A new hybrid projection iterative scheme is introduced to approximate a common element of the solution set of a generalized mixed equilibrium problem, the solution set of a variational inequality problem, and the set of fixed points of a relatively weak nonexpansive mapping in the Banach spaces. The obtained results generalize and improve the recent results announced by many other authors.     

Hybrid projection method for generalized mixed equilibrium problems, variational inequality problems, and fixed point problems in Banach spaces

A new hybrid projection iterative scheme is introduced to approximate a common element of the solution set of a generalized mixed equilibrium problem, the solution set of a variational inequality problem, and the set of fixed points of a relatively weak nonexpansive mapping in the Banach spaces. The obtained results generalize and improve the recent results announced by many other authors.     

Algorithms of common solutions to quasi variational inclusion and fixed point problems

The purpose of this paper is to present an iterative scheme for finding a common element of the set of solutions to the variational inclusion problem with multivalued maximal monotone mapping and inverse-strongly monotone mappings and the set of fixed points of nonexpansive mappings in Hilbert space.Under suitable conditions, some strong convergence theorems for approximating this common elements are proved. The results presented in the paper not only improve and extend the main results in Korpelevich（Ekonomika i Matematicheskie Metody,1976,12（4）：747-756）,but also extend and replenish the corresponding results obtained by Iiduka and Takahashi（Nonlinear Anal TMA,2005,61（3）：341-350）,Takahashi and Toyoda（J Optim Theory Appl,2003, 118（2）：417-428）,Nadezhkina and Takahashi（J Optim Theory Appl,2006,128（1）：191- 201）,and Zeng and Yao（Taiwanese Journal of Mathematics,2006,10（5）：1293-1303）.     

CONTACT PROBLEMS AND DUAL VARIATIONAL INEQUALITY OF 2－D ELASTOPLASTIC BEAM THEORY

ＣＯＮＴＡＣＴＰＲＯＢＬＥＭＳＡＮＤＤＵＡＬＶＡＲＩＡＴＩＯＮＡＬＩＮＥＱＵＡＬＩＴＹＯＦ２－ＤＥＬＡＳＴＯＰＬＡＳＴＩＣＢＥＡＭＴＨＥＯＲＹＤａｖｉｄＹａｎｇＧａｏ（高扬）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ．ＶｉｒｇｉｎｉａＰｏｌｙｔ...     

Several results and real problems in the theory of periodic motions and nonlinear mechanics (review)

Institute of Mathematics, Academy of Sciences of the Ukrainian SSR, Kiev. Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 24, No. 3, pp. 3–14, March, 1988.     

Degree theory for multivalued (S) type mappings and fixed point theorems

The main purpose of this paper is devoted to generalizing the results of Browder ^[1,2]. This paper consists of four parts. In the first part, we introduce the concepts of multivalued (S) and (S) type mappings and the concepts of the limits of multivalued (S) and (S) type mappings. These kinds of mappings contain many monotone type mappings, such as maximal monotone mapping, bounded pseudo-monotone mapping and bounded generalized pseudo-monotone mapping, as its special cases. In the second part we define the pseudo-degree for (S) type mapping and the degree for (S) ₊ type mapping. These two kinds of degrees are all the generalizations of the degree defined by Browder ^[1,2].As applications, we utilize the degree theory presented in part 2 to study the existence of solutions for the multivalued operator equations (see part 3) and to obtain some new fixed point theorems in part 4.Research Supported by the National Natural Science Founation of China.     

AN ITERATIVE METHOD FOR THE DISCRETE PROBLEMS OF A CLASS OF ELLIPTICAL VARIATIONAL INEQUALITIES

ＡＮＩＴＥＲＡＴＩＶＥＭＥＴＨＯＤＦＯＲＴＨＥＤＩＳＣＲＥＴＥＰＲＯＢＬＥＭＳＯＦＡＣＬＡＳＳＯＦＥＬＬＩＰＴＩＣＡＬＶＡＲＩＡＴＩＯＮＡＬＩＮＥＱＵＡＬＩＴＩＥＳＺｈｅｎｇＴｉｅ－ｓｈｅｎｓ（郑铁生）ＬｉＬｉ（李立）ＸｕＱｉｎｇ－ｙｕ（许庆余）（Ｄ...